§13.8 Asymptotic Approximations for Large Parameters

If $b\to \mathrm{\infty }$ in $ℂ$ in such a way that $\left|b+n\right|\ge \delta >0$ for all $n=0,1,2,\mathrm{\dots }$, then

For fixed $a$ and $z$ in $ℂ$

as $b\to \mathrm{\infty }$ in $|\mathrm{ph}b|\le \pi -\delta$, where $q_0(z,a)=1$ and

When the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when $|b|$ is large, and $|b-a|$ and $|z|$ are bounded.

Let $\lambda =z/b>0$ and $\zeta =\sqrt{2(\lambda -1-\ln \lambda )}$ with $\mathrm{sign}\left(\zeta \right)=\mathrm{sign}\left(\lambda -1\right)$. Then

and

as $b\to \mathrm{\infty }$, uniformly in compact $\lambda$-intervals of $(0,\mathrm{\infty })$ and compact real $a$-intervals. For the parabolic cylinder function $U$ see §12.2, and for an extension to an asymptotic expansion see Temme (1978).

Special cases are

and

To obtain approximations for $M(a,b,z)$ and $U(a,b,z)$ that hold as $b\to \mathrm{\infty }$, with $a>\frac{1}{2}-b$ and $z>0$ combine (13.14.4), (13.14.5) with §13.20(i).

Also, more complicated—but more powerful—uniform asymptotic approximations can be obtained by combining (13.14.4), (13.14.5) with §§13.20(iii) and 13.20(iv).

For other asymptotic expansions for large $b$ and $z$ see López and Pagola (2010).

For more asymptotic expansions for the cases $b\to \pm \mathrm{\infty }$ see Temme (2015, §§10.4 and 22.5)

For the notation see §§10.2(ii), 10.25(ii), and 2.8(iv).

When $a\to +\mathrm{\infty }$ with $b$ ($\le 1$) fixed,

where $w=\mathrm{arccosh}\left(1+{(2a)}^{-1}x\right)$, and $\beta =(w+\sinh w)/2$. (13.8.8) holds uniformly with respect to $x\in [0,\mathrm{\infty })$. For the case $b>1$ the transformation (13.2.40) can be used.

For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i).

When $a\to -\mathrm{\infty }$ with $b$ ($\ge 1$) fixed,

and

uniformly with respect to bounded positive values of $x$ in each case.

For asymptotic approximations to $M(a,b,x)$ and $U(a,b,x)$ as $a\to -\mathrm{\infty }$ that hold uniformly with respect to $x\in (0,\mathrm{\infty })$ and bounded positive values of $(b-1)/\left|a\right|$, combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii).

When $a\to \mathrm{\infty }$ in $\left|\mathrm{ph}a\right|\le \pi -\delta$ and $b$ and $z$ fixed,

where $C_{\nu }(a,\zeta )=\cos \left(\pi a\right)J_{\nu }\left(\zeta \right)+\sin \left(\pi a\right)Y_{\nu }\left(\zeta \right)$ and

where $c_0(z)=1$ and

For the Bernoulli numbers $B_k$ see §24.2(i) and for proofs and similar results in which $z$ can also be unbounded see Temme (2015, Chapters 10 and 27)

When $a,b\to +\mathrm{\infty }$ with $\left|z\right|$ and $\nu =\frac{a}{b}$ bounded

where ${\mathrm{\Gamma }}^{\ast }\left(a\right)$ is the scaled gamma function defined in (5.11.3). These results follow from Temme (2022), which can also be used to obtain more terms in the expansions. For generalizations in which $z$ is also allowed to be large see Temme and Veling (2022).